Combating Dissipation

7/29/2019 Combating Dissipation

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Combating Dissipation


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Numerical Dissipation

There are several sources of numerical

dissipation in these simulation methods

Error in advection step

Pressure projection (time splitting)

Not addressed yet in graphics!

Level set redistancing

Focus on the first


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Dissipation Example (1)

Start with a function nicely sampled

on a grid:


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The function moves to the left

(perfect advection) and is resampled


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And now we interpolate from new

sample values, and ruin it all!


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The Symptoms

For velocity:

Too viscous or sticky (molasses), or at animplausible length scale (scale model)

Turbulent detail quickly blurred away For smoke concentration:

Smoke diffuses into thin air too fast,nice sharp profiles or thin features vanish

For level sets:

Water evaporates into thin air, bubblesdisappear


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High Order/Resolution Schemes

That said, we can do a lot better thanfirst-order semi-Lagrangian

High order methods: use more data points

to get more accurate interpolation

Cancel out more terms in Taylor series

Problem: inevitably can give

undershoot/overshoot (too aggressive) Stability for nonlinear problems?

High resolution methods: high order except

near sharp regions


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Sharpening semi-Lagrangian

Can also do better with semi-Lagrangianapproach

Sharper interpolation

- e.g. limited Catmull-Rom [Fedkiw et al 02]

Estimating error and subtracting it

BFECC [e.g. Kim et al 05]

Using derivative information

CIP [e.g. Yabe et al. 01]


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Example

Exact (particles) vs. 1st order vs. BFECC


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Aside: resampling

Closely related to the sampling theorem:

frequencies above a certain limit cannot be

reliably recovered on a grid

Sharp features have infinitely high

frequency!

Schemes which use an Eulerian grid as

fundamental structure are inherently limited

(forced to use higher resolution than is

strictly necessary)


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Particle-in-Cell Methods

Back to Harlow, 1950s, compressible flow

Abbreviated PIC

Idea:

Particles handle advection trivially

Grids handle interactions efficiently

Put the two together:

- transfer quantities to grid- solve on grid (interaction forces)- transfer back to particles- move particles (advection)


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Start with particles

Transfer to grid

Resolve forces on grid

Gravity, boundaries,

pressure, etc.

Transfer velocity back toparticles

Advect: move particles

PIC


Transfer to grid



pressure, etc.




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Transfer to grid



pressure, etc.



PIC


Transfer to grid



pressure, etc.




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Transfer to grid



pressure, etc.



PIC


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Transfer to grid



pressure, etc.



PIC


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Transfer to grid



pressure, etc.



PIC


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FLuid-Implicit-Particle (FLIP)

Problem with PIC: we resample (average) twice

Even more numerical dissipation than pureEulerian methods!

FLuid-Implicit-Particle (FLIP) [Brackbill & Ruppel86]:

Transfer back the change of a quantity fromgrid to particles, not the quantity itself

Each delta only averaged once: noaccumulating dissipation!

Nearly eliminated numerical dissipation fromcompressible flow simulation

Incompressible FLIP [Zhu&Bridson05]


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Wheres the Catch?

Accuracy:

When we average from particles to grid, simpleweighted averages is only first order

Not good enough for level sets

Noise:

Typically use 8 particles per grid cell for decentsampling

Thus more degrees of freedom in particles then grid The grid simulation cant see/respond to small-scale

particle variations can potentially grow in time

Regularize: e.g. 95% FLIP, 5% PICCan actually determine ratio which matches a particularphysical viscosity!

Combating Dissipation

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